The rest of paper is organized as follows. In Section , we represent our techniques on Liu-Viswanath extremal inequality, and show how to prove Gaussian optimality without channel enhancement argument. In Section , we illustrate the similar idea can also recover the vector generalization of Costa's EPI. We further revisit the main result of [1]} and [2]} in Section  and , separately.
The fundamental limits of the two mulit-user information theory problems are fully characterized via our perturbation methods. Finally, we conclude by summarizing our contributions in the context of applications in mulit-user information theory problems.
